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Second and higher-order derivatives
Differentiate again to get acceleration, concavity, and the test that classifies turning points.
Differentiating twice gives the second derivative, written \(f''(x)\), \(\frac{d^{2}y}{dx^{2}}\) or \(y''\). It is the rate at which the rate is changing: if \(f\) is position and \(f'\) is velocity, then \(f''\) is acceleration. Set the Order control on the calculator above to 2 — or use the smaller one here — and the working shows every round.
What the second derivative tells you
- \(f'' > 0\): the curve is concave up, holding water, and \(f'\) is increasing.
- \(f'' < 0\): the curve is concave down, and \(f'\) is decreasing.
- \(f'' = 0\) with a sign change on either side: an inflection point, where the curvature flips.
The second derivative test
At a point where \(f'(a) = 0\), the second derivative classifies it without checking signs on both sides: \(f''(a) > 0\) means a minimum, \(f''(a) < 0\) means a maximum, and \(f''(a) = 0\) is inconclusive — fall back to the first derivative test there.
Third, fourth and nth derivatives
Keep going and the notation switches from primes to \(f^{(4)}\), \(f^{(5)}\), \(f^{(n)}\). Some functions settle into a pattern: \(e^{x}\) is unchanged at every order, \(\sin x\) cycles back to itself every four, and a polynomial of degree \(n\) reaches zero at order \(n+1\). Others grow without ever repeating — a fifth-order derivative of a product can run to a hundred steps, which is why the working is written out a batch at a time.
Cara kerja Derivative Calculator
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Penguraian
Teks Anda menjadi pohon operasi, dan tanda perkalian yang dihilangkan ditambahkan secara otomatis.
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Penataan
Pohon itu ditampilkan dalam notasi matematika sehingga Anda dapat memeriksa bahwa kalkulator membaca yang Anda maksud.
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Penurunan
Satu aturan per simpul — jumlah, hasil kali, hasil bagi, pangkat, rantai — masing-masing dicatat sebagai langkah.
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Penyederhanaan dan grafik
Hasilnya disederhanakan dengan aritmetika eksak, lalu kedua kurva digambar.
Notasi dan aturan
Semua yang diterima parser dan aturan yang diterapkan mesin. Setiap nama aturan menuju halamannya sendiri.
Aturan yang dipakai
| Aturan | Pernyataan |
|---|---|
| Constant | \(\frac{d}{dx}c = 0\) |
| Variable | \(\frac{d}{dx}x = 1\) |
| Sum | \(\left(f + g\right)' = f' + g'\) |
| Constant multiple | \(\left(cf\right)' = c\,f'\) |
| Power | \(\frac{d}{dx}x^{n} = n x^{n-1}\) |
| Product | \(\left(fg\right)' = f'g + fg'\) |
| Quotient | \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^{2}}\) |
| Chain | \(\left(f \circ g\right)' = f'(g)\,g'\) |
| Exponential | \(\frac{d}{dx}a^{x} = a^{x}\ln a\) |
| Logarithm | \(\frac{d}{dx}\ln x = \frac{1}{x}\) |
| Logarithm, base a | \(\frac{d}{dx}\log_{a} x = \frac{1}{x\ln a}\) |
| Trigonometric | \(\frac{d}{dx}\sin x = \cos x\) |
| Inverse trig | \(\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}\) |
| Hyperbolic | \(\frac{d}{dx}\tanh x = \operatorname{sech}^{2} x\) |
| Inverse hyperbolic | \(\frac{d}{dx}\operatorname{artanh} x = \frac{1}{1 - x^{2}}\) |
| Absolute value | \(\frac{d}{dx}\left|x\right| = \operatorname{sgn} x, \; x \neq 0\) |
| Logarithmic | \(\frac{d}{dx}f^{g} = f^{g}\left(g'\ln f + \frac{gf'}{f}\right)\) |
| Rewrite in natural logs | \(\log_{a} b = \frac{\ln b}{\ln a}\) |
Sintaks masukan
| Anda ketik | Artinya |
|---|---|
2x, 3(x+1) | Implicit multiplication |
x^3, x**3 | Powers — x² and x³ also work |
sqrt(x), √x | Square root |
cbrt(x) | Cube root |
e^x, exp(x) | Exponential function |
ln(x), log(x) | Natural logarithm |
log(2, x), log2(x) | Logarithm to a given base |
sin^2(x) | The square of sin x |
tan^-1(x), arctan(x) | Inverse tangent |
|x-1|, abs(x-1) | Absolute value |
pi, e, tau | Constants |
x_1, theta | Subscripted and Greek variables |
More derivative tools
The calculator above handles ordinary derivatives to any order. These pages take the same engine into the cases that need their own controls or their own explanation.
Partial derivatives
Work with two or three variables and differentiate with respect to any one of them.
Derivative at a point
Get the slope as a number at any x, and the equation of the tangent line there.
Derivative graphs
Plot a function against its derivative and read the turning points straight off.
Implicit differentiation
Find dy/dx for curves such as x² + y² = 25 that are not solved for y.
Parametric and polar
Curves given by x(t), y(t) or by r(θ), where dy/dx comes from the parameter.
Directional derivatives
The gradient of a surface, and how steeply it rises in any direction you choose.
Numerical differentiation
Estimate a derivative from values alone, and compare it with the exact answer.
Derivative of an integral
The Fundamental Theorem, and the chain rule for a variable limit.
Learn the mathematics
Explanations rather than tools, for when the answer is not the point.
What is a derivative?
The definition, the notation, and where derivatives sit in each calculus course.
First principles
The limit definition worked slowly, for x², √x and sin x.
Practice problems
Problems grouped by rule, with answers and the standard derivatives table.
Formula cheat sheet
Every rule and standard derivative on one printable page.
Looking for a specific rule? Aturan lists all eighteen, each with worked examples and a graph.
Pertanyaan
Apakah kalkulator menampilkan setiap langkah?
Ya. Setiap aturan yang diterapkan dicatat berurutan, disertai hukum umum yang dipakai dan penjelasan singkat. Ekspresi panjang ditampilkan bertahap agar halaman tetap cepat.
Fungsi apa saja yang didukung?
Polinomial dan akar, eksponen dan logaritma dengan basis apa pun, keenam fungsi trigonometri beserta inversnya, keluarga hiperbolik beserta inversnya, ditambah nilai mutlak dan fungsi tanda.
Bisakah menghitung turunan tingkat tinggi?
Bisa, sampai tingkat sepuluh. Setiap putaran diturunkan penuh dan langkah rincinya ditampilkan untuk putaran pertama.
Apakah masukan saya dikirim ke server?
Tidak. Penguraian, penurunan, penyederhanaan, dan penggambaran grafik semuanya terjadi di peramban Anda. Ekspresi terbaru hanya tersimpan di perangkat Anda.
Bisakah dipakai di ponsel?
Bisa. Tata letak menjadi satu kolom, grafik menerima seretan dan zum dua jari, dan papan tik fungsi menyediakan simbol yang merepotkan diketik di ponsel.